arXiv research

Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

Classifies holomorphic parabolic geometries on complex manifolds.

Extends parabolic study to flat hyperkähler manifolds.

Sharp estimates for parabolic equations on manifolds using symmetrization.

The paper explores automorphism groups of parabolic structures on aspherical manifolds.

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify th…

The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.

We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.

The paper defines capacities for minimal graphs over manifolds and proves the half-space property.

We prove that any simply connected special Kaehler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. Also we show that a classical result of Calabi and Pogorelov on parabolic spheres implies …

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.

In the paper we prove that every closed orientable three-manifold admits a parabolic foliation.

Study of BGG sequences on foliated manifolds with transverse parabolic geometry.

The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.

Study rigid classes on hyperkahler manifolds, showing general ones are rigid.

Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.

We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth c…

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The…

Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

Study proves correspondence for special bundles on complex surfaces.

Study proves long-term solutions to a specific equation on hyperKähler manifolds.

Sharp bounds and parabolicity results for 3-manifolds with scalar curvature.

Sharp heat kernel estimates on manifolds lead to solutions of the Parabolic Anderson model.

Study of parabolic-preserving deformations of hyperbolic lattices.

Solves long-time solutions for a specific equation on hyperkähler manifolds.

We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization procedure and a fixed-point argument, and the key ingredient is the well known Sch…

In this article we establish a local parabolic almost monotonicity formula for two phase free boundary problems on Riemannian manifolds, which is an extension of a work of Edquist-Petrosyan.

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of t…

In this short note we extend Chow and Lu's advanced maximum principles for parabolic systems on closed manifolds to the case of compact manifolds with boundary, which also generalizes a Hopf type theorem of Pulemotov.

Generalized Agol's theorem to 3-manifold groups.

Survey on Chern-Ricci flow for complex manifolds.

We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and pa…

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

Study shows long-term solutions for complex equations on curved spaces.

For a semisimple Lie group $G$ with parabolic subgroups $Q\subset P\subset G$, we associate to a parabolic geometry of type $(G,P)$ on a smooth manifold $N$ the correspondence space $\Cal CN$, which is the total space of a fiber bundle over $N$ with fiber a generalized flag manifold, and construct a canonical parabolic…

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

The paper establishes Harnack inequalities for solutions of nonlinear parabolic equations on manifolds with integral Ricci curvature bounds.

Let M be a compact, holomorphically symplectic Kahler manifold, and $η$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $η$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $η$ are coisotropic. When M is generic, this is used to show that all Le…

This paper studies parabolic quasi-Coxeter elements in complex reflection groups and their combinatorial properties.

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Zadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manif…

We consider rank 3 distributions with growth vector (3,5,6). The class of such distributions splits into three subclasses: parabolic, hyperbolic and elliptic. In the present paper, we deal with the parabolic case. We provide a classification of such distributions and exhibit connections between them and Gl(2)-structure…